 Low-pass filter of a multiresolution analysis

For an orthonormal scaling function satisfying $\phi (x)=\sqrt{2}{\sum }_k{h}_k\phi (2x-k)$, its low-pass Fourier series symbol is $m(\xi )=2^{-1/2}{\sum }_k{h}_k{e}^{-ik\xi }$. The scaling refinement equation becomes $\hat{\phi }(2\xi )=m(\xi )\hat{\phi }(\xi )$. The symbol is initially an almost-everywhere defined periodic function; the assumption that the scaling function is Lebesgue integrable alone does not make it smooth. The orthonormal translates give the quadrature mirror filter identity.